Principal Trotter observation error with truncated commutators

Hamiltonian simulation is one of the most promising applications of quantum computers, and the product formula is one of the most important methods for this purpose. Previous related work has mainly focused on the worst-case, average-case, or actual-case scenarios. In this work we consider the simulation error under a fixed observable. Specifically, errors that commute with this observable become less significant. To illustrate this point, we define the observation error as the expectation under the observable and provide a commutativity-based upper bound using the Baker-Campbell-Hausdorff formula. Furthermore, we observe that the evolution order plays a critical role in determining the observation error. By leveraging a simulated annealing algorithm, we design an evolution order optimization algorithm, to identify an order that enhances commutativity with the observable, where the observation error indicated by this upper bound can be significantly compressed. In the experiment with the Heisenberg model, the observation bound compresses the Trotter number by nearly half compared to recent commutator bounds. The experiment on the hydrogen molecule Hamiltonian demonstrates that optimizing the order can lead to nearly half the reduction in the Trotter number.